1. Field of the Invention
The present invention relates to a method for automatically aligning an astronomical telescope, specifically to a method for automatically aligning a telescope with feedback control. A motor speed is controlled so as to track a bright star and the coordinates of the telescope.
2. Description of Prior Art
The celestial bodies can be represented by a spherical polar coordinate on the sky. FIG. 1 shows the schematic diagram of a spherical polar coordinate with polar axis OZ, wherein the reference plane is the XOA plane vertical to the polar axis OZ and OX is a reference direction on the reference plane. The projection of OP on the reference plane is indicated by OA. Therefore, a celestial body P can be represented by the angle XOA and the angle AOP, as shown in FIG. 1.
The most common sphere coordinates include celestial sphere coordinates, and altitude and azimuth coordinates, which are described below.
The Celestial Sphere Coordinate
In the celestial sphere coordinate, the observer on earth is located at sphere center O and the polar axis OZ is an axis parallel to the rotation axis O′Z′ of the Earth and the position direction is the direction of north celestial pole. The reference plane is a plane passing the center O and vertical to the polar axis. The reference direction OX on the reference plane is the intersecting line of the celestial meridian plane of the location of the observer on the Earth and the reference plane. The positive direction is the direction pointing from the rotation axis of the Earth toward the observer.
The Altitude and Azimuth Coordinate
In the altitude and azimuth coordinate, the observer is located at the center O and the reference plane XOA is a horizontal plane (the tangential plane to the surface of the Earth) normal to the observer. The reference direction OX on the reference plane is south and the positive direction of polar axis OZ is to the Zenith.
The Telescope Mount, the Telescope Coordinate System and Az-Alt Telescope Mount
In general, the telescope mount is a mechanical system defined by two orthogonal rotational axes. The pointing direction of the telescope can be represented by the angle formed by two of the rotational axes. This is also a spherical coordinate system and is referred to as telescope coordinate system.
Provided that a rotational axis of the telescope mount is coincidental with a polar axis of the altitude and azimuth coordinate, the telescope mount is referred to as an Alt-Az telescope mount. The rotational axis pointing to the Zenith is referred to as the Az (Azimuth)axis, the rotational axis parallel to the ground level is referred to as the Alt (Altitude) axis. The reference plane of the Alt-Az telescope mount is parallel to the horizontal plane.
The Representation of Celestial Objects in Different Coordinates
The coordinate of a celestial object in celestial sphere coordinate can be transformed to the altitude and azimuth coordinates (or the telescope mount coordinate system) by coordinate transform, as shown to FIG. 1.
In celestial sphere coordinates, angle XOA=α and angle AOP=β;
In altitude and azimuth coordinates, angle XOA=α′ and angle AOP=β′;
the angles have following relationship:
      [                                        cos            ⁢                                                  ⁢            α            ⁢                                                  ⁢            cos            ⁢                                                  ⁢            β                                                            sin            ⁢                                                  ⁢            α            ⁢                                                  ⁢            cos            ⁢                                                  ⁢            β                                                            sin            ⁢                                                  ⁢            β                                ]    =            [      M      ]        ×          [                                                  cos              ⁢                                                          ⁢                              α                ′                            ⁢                                                          ⁢              cos              ⁢                                                          ⁢                              β                ′                                                                                        sin              ⁢                                                          ⁢                              α                ′                            ⁢                                                          ⁢              cos              ⁢                                                          ⁢                              β                ′                                                                                        sin              ⁢                                                          ⁢                              β                ′                                                        ]      GOTO Telescope
The GOTO telescope is a motor driven and computer-based telescope. One basic function of the GOTO telescope is to convert a coordinate A(α, β) in celestial sphere coordinates to a coordinate A′(α′, β′) in telescope mount coordinates. The mount is controlled to move from an initial arbitrary position to the coordinate A′(equivalent to the coordinate A), whereby the celestial object at coordinate A appears on field of view in the telescope.
Alignment of GOTO Telescope
To precisely locate the target, the GOTO telescope requires alignment, which comprises following steps:
1. The GOTO telescope points to a first position in the sky and the celestial sphere coordinate (α1, β1) and the telescope mount coordinate (α1′, β1′) associated with the first position are recorded.
2. The GOTO telescope points to a second position in the sky and the celestial sphere coordinate (α2, β2) and the telescope mount coordinate (α2′, β2′) associated with the second position are recorded.
3. The GOTO telescope points to a third position in the sky and the celestial sphere coordinate (α3, β3) and the telescope mount coordinate (α3′, β3′) associated with the third position are recorded.
4. The conversion relationship between celestial sphere coordinates and telescope mount coordinates can be determined by following formula:
      [    M    ]    =            [                                                  cos              ⁢                                                          ⁢              α1              ⁢                                                          ⁢              cos              ⁢                                                          ⁢              β1                                                          cos              ⁢                                                          ⁢              α2cos              ⁢                                                          ⁢              β2                                                          cos              ⁢                                                          ⁢              α3              ⁢                                                          ⁢              cos              ⁢                                                          ⁢              β              ⁢                                                          ⁢              3                                                                          sin              ⁢                                                          ⁢              α              ⁢                                                          ⁢              1              ⁢              cos              ⁢                                                          ⁢              β1                                                          sin              ⁢                                                          ⁢              α              ⁢                                                          ⁢              2              ⁢              cos              ⁢                                                          ⁢              β2                                                          sin              ⁢                                                          ⁢              α3cos              ⁢                                                          ⁢              β3                                                                          sin              ⁢                                                          ⁢              β1                                                          sin              ⁢                                                          ⁢              β2                                                          sin              ⁢                                                          ⁢              β3                                          ]        ×                  [                                                            cos                ⁢                                                                  ⁢                                  α1                  ′                                ⁢                cos                ⁢                                                                  ⁢                                  β1                  ′                                                                                    cos                ⁢                                                                  ⁢                                  α2                  ′                                ⁢                cos                ⁢                                                                  ⁢                                  β2                  ′                                                                                    cos                ⁢                                                                  ⁢                                  α3                  ′                                ⁢                cos                ⁢                                                                  ⁢                                  β3                  ′                                                                                                        sin                ⁢                                                                  ⁢                                  α1                  ′                                ⁢                cos                ⁢                                                                  ⁢                                  β1                  ′                                                                                    sin                ⁢                                                                  ⁢                                  α2                  ′                                ⁢                cos                ⁢                                                                  ⁢                                  β2                  ′                                                                                    sin                ⁢                                                                  ⁢                                  α3                  ′                                ⁢                cos                ⁢                                                                  ⁢                                  β3                  ′                                                                                                        sin                ⁢                                                                  ⁢                                  β1                  ′                                                                                    sin                ⁢                                                                  ⁢                                  β2                  ′                                                                                    sin                ⁢                                                                  ⁢                                  β3                  ′                                                                    ]                    -        1            where the telescope mount coordinates (α1′, β1′), (α2′, β2′), (α3′, β3′) can be obtained through the motor control system of the mount; and the method for obtaining the celestial sphere coordinates (α1, β1), (α2, β2), (α3, β3) will be detailed below.How to Obtain the Celestial Sphere Coordinates
Most stars in the sky are documented by relative positions in a celestial objects database, wherein the coordinates of a celestial body in the database are represented by right ascension (RA) and declination (DEC) and RA and DEC are angular coordinates.
Provided that the date and time T0 at zero-degree longitude is known, the Sidereal Time at Greenwich meridian, labeled as LST0, can be calculated and the calculation is omitted here for clarity.
Provided that the longitude of user location is L, the celestial object with known (Ra, Dec) can be expressed in term of HA (hour angle, which is equivalent to angle XOA in FIG. 1) and Dec (equivalent to angle AOP in FIG. 1) with reference to the celestial sphere coordinates of the observer. The hour angle HA can be calculated according to the following formula:HA=LST0−L−Ra
Therefore, the alignment of the GOTO telescope is first performed by pointing the GOTO telescope to three stars, which are referred to as alignment stars. The corresponding RA and DEC data for those alignment stars can be obtained from a database and the celestial sphere coordinates (α1, β1), (α2, β2), (α3, β3) for the telescope can be determined using the longitude and the current local time of the observer.
Difficulty in Identifying Alignment Star
As can be seen in above description, one essential step in aligning the GOTO telescope is pointing the GOTO telescope at an alignment star. However, an error will occur if the coordinates for star A in the database are used, but the actual star pointed at by the GOTO telescope is star B.
Unfortunately, for novice astronomer, this error can happen easily, which might be caused by their unfamiliarity with name or number of the alignment star or erroneous pointing of the GOTO telescope.